L-optimal transportation for ricci flow

Transcription

1 L-optimal transportation for ricci flow Peter Topping Abstract We introduce the notion of L-optimal transportation, and use it to construct a natural monotonic quantity for Ricci flow which includes a selection of other monotonicity results, including some key discoveries of Perelman [13] both related to entropy and to L-length and a recent result of ccann and the author [11]. 1 Introduction Given a closed manifold, of dimension n, a smooth family gt of Riemannian metrics is called a Ricci flow if it satisifes the nonlinear PDE g t introduced by Hamilton [8] see [16] for further information. = Ricgt, 1.1 In order to do analysis on Ricci flows, one has been traditionally reliant largely on the maximum principle. In particular, one does not have a Sobolev inequality; more precisely one has no a priori control on the evolution of the standard Sobolev constant. Instead, one can look for other quantities which are controlled under Ricci flow, the best known of which is the optimal constant in a certain log-sobolev inequality. That log-sobolev constant is monotonic in time by virtue of the monotonicity of Perelman s W entropy see [13] and [16] for details, and its application to proving no local collapsing for Ricci flow. The goal of this paper is to introduce a new geometric quantity for Ricci flow which is also monotonic, and which simultaneously generalises Perelman s W entropy and one of Perelman s crucial monotonicity results involving his celebrated notion of L-length. Furthermore, the monotonicity of our new quantity includes a recent result of ccann and the author [11] where Ricci flow was considered in conjunction with the theory of optimal transportation. The new quantity elucidates why these previous entropies and other quantities function the way they do, and indicate the extent to which we can hope to generalise them to other geometric flows. To describe the new quantity, we introduce a new notion of optimal transportation of measures through space-time in Ricci flow, and an associated notion of Wasserstein-type November

2 distance between probability measures. Before we can describe this concept, we must first survey how one can make sense of a distance between two points in space-time. In light of the work of Perelman, it is convenient to consider the Ricci flow backwards in time. To this end, we adopt the notation τ to represent some backwards time parameter i.e. τ = C t for some C R and consider the reverse Ricci flow g τ = Ricgτ, defined on a time interval including [τ 1, τ ] where 0 τ 1 < τ. Perelman s L-length of a path γ : [τ 1, τ ] where one should view the point γτ as a point in the Riemannian manifold, gτ is defined [13] by τ Lγ := τ Rγτ, τ + γ τ gτ dτ, 1. τ 1 where Rx, τ is the scalar curvature at x in, gτ. One can use such a length to give rise to a distance, mirroring the classical construction of Riemannian geometry: We define the L-distance between a point x, τ 1 and y, τ where x, y and 0 τ 1 < τ are times as Qx, τ 1 ; y, τ := inf{lγ γ : [τ 1, τ ] is smooth and γτ 1 = x, γτ = y}, with the caveat that this distance can be negative, and one is not directly generating a metric space via this construction. When τ 1 and τ are pushed together, the scalar curvature term in the definition 1. of L is dwarfed by the energy term, and one recovers the classical Riemannian distance in the sense that lim τ τ 1 Qx, τ 1 ; y, τ = d x, y, τ 1, 1.3 τ τ 1 uniformly in x and y, where d,, τ is the Riemannian distance with respect to gτ. Equipped with Q, we can introduce the L-Wasserstein distance V ν 1, τ 1 ; ν, τ between two Borel probability measures ν 1 and ν, viewed at times τ 1 and τ respectively: V ν 1, τ 1 ; ν, τ := Qx, τ 1 ; y, τ dπx, y 1.4 inf π Γν 1,ν where Γν 1, ν is the space of Borel probability measures on with marginals ν 1 and ν i.e. πω = ν 1 Ω and π Ω = ν Ω for Borel Ω. By virtue of 1.3, we can recover the standard -Wasserstein distance W from V in the limit that τ τ 1 : lim τ τ 1 V ν 1, τ 1 ; ν, τ = W ν 1, ν, τ 1 τ τ 1 := inf π Γν 1,ν d x, y, τ 1 dπx, y. 1.5 Whilst the distance V will be the main ingredient of our new result, all of the results we will discuss in this paper are phrased or can be rephrased in terms of the probability densities of Brownian diffusion on Ricci flows, backwards in time that is, forwards in τ. In other words, we consider families ντ of Borel probability measures so that if τ a < τ b and ντ a represents the probability of the location of a Brownian particle at time τ a, then ντ b represents the probability of the location of the particle at time τ b. athematically, if we denote the Riemannian volume measure on, gτ by µτ, and write dντ = uτdµτ for some evolving probability density u : τ a, τ b 0, then u satisfies the equation u = u Ru, 1.6 τ

3 where the scalar curvature term is arising because of the evolution of the volume element τ dµτ = 1 g tr τ dµτ = Rdµτ see [16,.5.7]. By considering families ντ over open intervals, we may always assume that ν is a smooth family of positive measures, by which we mean that its density u is smooth and strictly positive. For brevity, throughout the paper we will refer to such families ντ satisfying 1.6 simply as diffusions. It is a general principle which can be extracted from Perelman s work [13] that the properties of such diffusions are related to the properties of the Ricci flow itself. This mirrors the classical connection between the geometry of fixed Riemannian manifolds and the properties of the heat kernels they support. Our main theorem asserts the monotonicity of a renormalised version of the L-Wasserstein distance between two diffusions, at different times. The quantity has a global space-time aspect, but localising or restricting it will reveal some more familiar monotonic quantities. Theorem 1.1. Suppose that 0 < τ 1 < τ and gτ is a reverse Ricci flow on a closed manifold of dimension n, for τ in some open interval containing [ τ 1, τ ]. Suppose that ν 1 τ and ν τ are two diffusions as defined above for τ in some neighbourhoods of τ 1 and τ respectively. Let τ 1 = τ 1 s := τ 1 e s, τ = τ s := τ e s be two exponential functions of s R, and define the renormalised distance between the diffusions ν 1 and ν at s by Θs := τ τ 1 V ν 1 τ 1, τ 1 ; ν τ, τ n τ τ 1 for s in a neighbourhood of 0 such that ν i τ i s are defined i = 1,. Then Θs is a weakly decreasing function of s. The fact that we should track the diffusions ν 1 and ν with this exponential parametrisation is somewhat unconventional but is natural when one considers the invariance of Ricci flow under parabolic rescaling [16, 1..3]. We will prove Theorem 1.1 in Section 4. Before doing so, we will have to develop the theory of L-optimal transportation Section in order to understand the structure of the minimiser π Γν 1, ν which will exist for the variational problem in 1.4. This will lead us to a construction of what we will call L-Wasserstein geodesics between two given probability measures. In Section 3 we will investigate the properties of the classical Boltzmann-Shannon entropy along these L-Wasserstein geodesics. This will involve investigating carefully the behaviour of L-geodesics for Ricci flow, and their L-Jacobi fields, and making natural computations for the second derivatives of the volume element along L-geodesics which extend the first derivative calculations which were used so successfully by Perelman [13]. Luckily, many of the optimal transportation aspects of this theory can be developed along similar lines to the the development of the original rigorous theory of optimal transportation on Riemannian manifolds. In particular, we follow the work of ccann [10] and Cordero-Erausquin, ccann and Schmuckenschläger [6] wherever possible. Heuristics which motivated some of that original theory can be found in work of Otto and Villani [1]. Cedric Villani has pointed out to us that an alternative to developing the optimal transport structure theory following [10] and [6] would be to invoke the theory applicable to very general cost functions which is developed in his forthcoming lecture notes [18]. Optimal transport in this generality is also considered in [3] as pointed out to us by Robert ccann. The proof of our main result itself is closest in spirit and detail to our previous work [11] with ccann. Before proceeding with this detail, we explain how the results of Perelman, and ccann and the author, fall naturally out of Theorem 1.1, as alluded to earlier. We are only 3

4 looking for quantities which are adapted to studying shrinking solitons in Ricci flow; modifications to the theory could be made to recover corresponding quantities adapted to steady or expanding solitons if they were required. 1.1 Recovering the result of ccann-topping We have seen in 1.5 how the standard Wasserstein distance arises W in a limit of our L-Wasserstein distance V. In Lemma B.1 and Corollary B.3 of Appendix B, we will sharpen this relationship. Turning to Theorem 1.1, if we take τ τ 1, then for each s, Θs W ν 1 τ 1, ν τ 1, τ 1, and we find: Corollary 1.. ccann-topping [11] Given two diffusions ν 1 τ and ν τ as defined earlier on a reverse Ricci flow gτ, the function is weakly decreasing in τ. τ W ν 1 τ, ν τ, τ This result leads in [11] to a characterisation of supersolutions to the Ricci flow equation, which can be exploited to give a notion of weak solutions for Ricci flow. 1. Recovering Perelman s W-entropy Perelman s celebrated W-entropy is used to prove no local collapsing for Ricci flow, and it lies behind Perelman s pseudolocality result [13]. To recover it, we need also to consider the limit as τ 1 and τ approach each other. However now, we consider the case that ν 1 τ and ν τ coincide. By the previous case, our renormalised distance Θs will be zero in the limit τ τ 1, so in this case, we will look at the next term in the expansion of Θs in terms of τ τ 1 to get a new monotonic quantity. We will need to consider the infinitesimal version of the L-Wasserstein distance implied in the following lemma. Given a smooth family of positive probability measures ντ on a closed manifold, for τ in some neighbourhood of τ 1, we call a vector field X ΓT an advection field for ντ at τ = τ 1 if there exists a smooth family of diffeomorphisms ψ τ :, for τ in a neighbourhood of τ 1, with ψ τ1 the identity, and such that ψ τ # ντ 1 = ντ and X = ψ τ τ=τ 1. Lemma 1.3. Suppose gτ is a reverse Ricci flow, and ντ is a smooth family of positive probability measures on a closed manifold, for τ in some neighbourhood of τ 1 R. Then as τ τ 1, [ V ντ 1, τ 1 ; ντ, τ = τ τ 1 inf τ1 X + oτ τ 1, ] R, τ 1 + X gτ 1 dντ 1 where the infimum is taken over all advection fields X for ντ at τ = τ In this lemma, we are choosing the advection field X above to have the least kinetic energy ; the minimising X can be written explicitly as the gradient of v : R solving divu v = du dτ at τ = τ 1, where Uτ is the one-parameter family of probability 4

5 densities satisfying dντ = Uτdµτ 1 with µτ 1 representing the Riemannian volume measure for gτ 1. It is the coefficient of τ τ 1 in 1.7 the part within square brackets which we call the infinitesimal L-Wasserstein distance, or L-Wasserstein speed of ντ, with respect to gτ, at τ = τ 1. We delay the proof of Lemma 1.3 until Appendix B. Let us apply this lemma in the case of Theorem 1.1 specialised to the situation that ν 1 τ = ν τ; we will write this measure simply as ντ. We denote its probability density with respect to Riemannian volume measure µτ by uτ := dντ dµτ and its probability density with respect to µτ 1 by Uτ := dντ U dµτ 1 as before. Then τ = U = u at τ = τ 1, so the optimal advection field is given by X = ln u. Let us write τ = 1+η τ 1, so the functions τ 1 s and τ s of the theorem satisfy τ s = 1 + ητ 1 s for all s. In this situation, Lemma 1.3 tells us that Θs = η τ 1 +oη, where for each τ, R + ln u dντ 1 nτ 1 F = R + ln u u dµ +oη = η τ 1 Fτ 1 nτ 1 is Perelman s F-information functional [13], [16, 6.]. Theorem 1.1 then tells us that τ Fτ nτ is weakly decreasing in τ. 1.8 We are interested in understanding Perelman s W-entropy which is normally written for given τ as where f : R is defined by u = W = [τ f + R + f n ] u dµ, e f 4πτ n. See [13] and [16] for more information and applications to proving no local collapsing. Now a short calculation shows that dw dτ = 1 d τ Fτ nτ τ dτ so by 1.8 we recover the monotonicity of W: dw dτ Recovering Perelman s enlarged length monotonicity Whereas we have considered distances Qx, τ 1 ; y, τ so far, most of Perelman s constructions involve the special case Ly, τ := Qx, 0; y, τ for fixed x, or variants thereof. In particular, he defines the enlarged distance Ly, τ := τly, τ, and proves that the minimum over of L, τ nτ is a weakly decreasing function of τ. Because the minimum is zero in the limit τ 0, this implies that for any τ, one can always find a point y for which Ly, τ nτ, and that fact turns out to be essential in Perelman s arguments to extract asymptotic solitons for κ-solutions, and also to prove no local collapsing estimates when one is studying Ricci flows with surgery. See [13] and [14] for more details. Here we point out that the above monotonicity is also encoded in our Theorem 1.1. To see this, we would like to set τ 1 = 0. Strictly speaking, we have assumed that τ 1 > 0 5

6 to avoid dealing with a host of special cases and technical issues in the proofs; we leave the reader either to extend the theory, or take a limit τ 1 0. The exponential function τ 1 s will then be zero for all s. For ν 1 τ, we take the diffusion which at τ = 0 is the point unit mass δ x centred at x. Therefore ν 1 τ 1 s is that same measure for all s, and because the minimising π in the transportation problem defining V δ x, 0, ν τ, τ will be δ x ν τ, we have V δ x, 0, ν τ, τ = L, τ dντ, and hence Θs = L, τ nτ dν τ. Theorem 1.1 then shows that the function τ L, τ nτ dν τ is weakly decreasing, which because ν τ is an arbitrary diffusion, tells us that the minimum of the integrand is also weakly decreasing. 1.4 Fixed manifolds A further precursor to Theorem 1.1 is the work of Sturm and von Renesse [15]. They showed that on a fixed Riemannian manifold of weakly positive Ricci curvature, the Wasserstein distance between two diffusions is decreasing. Our results intersect in the special case that one considers a Ricci flat Riemannian manifold. Acknowledgements: I would like to thank Robert ccann for many useful conversations about optimal transportation. Following our previous work [11] with ccann, I would like to thank John Lott and Ben Chow for encouraging the search for links between the results in [11] and the entropy monotonicity and L-length theory in [13], respectively. Overview of L-optimal transportation Throughout this section, we will be considering a smooth reverse Ricci flow gτ defined on an open time interval including some interval [τ 1, τ ] with 0 < τ 1 < τ. Our goal is to understand the variational problem from 1.4. To begin, we note that Γν 1, ν from 1.4 is a weak- compact subset of the dual to the Banach space of continuous functions on equipped with the C 0 norm, and so we can be sure of the existence of a minimiser π Γν 1, ν 1 for the variational problem in 1.4 by the Banach-Alaoglu theorem. We call this minimising π the optimal transference plan, reusing the standard terminology from standard mass transportation theory. However, in order to rigorously prove anything about L-optimal transportation, we must understand the structure of the minimising π in some detail, and that is what we address now. In order to discuss these issues, we need some basic theory of Perelman s L-length. The more elaborate theory we require along these lines will be relegated to Appendix A. We have already introduced Perelman s notion of L-distance; he also introduced [13] a notion of L-geodesic γ : [τ 1, τ ] analogous to the usual Riemannian notion, which satisfies the equation D τ X = 1 1 R RcX τ X, where X = γ τ, Rc is the Ricci curvature 6

7 viewed as an endomorphism, and D τ represents the pull-back under γ of the Levi-Civita connection on, gτ, acting in the direction τ. This notion of geodesic then gives rise to an L-exponential map L τ1,τ exp x : T x which maps a vector Z T x to the point γτ, where γ : [τ 1, τ ] is the unique L-geodesic such that γτ 1 = x and τ 1 γ τ 1 = Z. Consider, for the moment, the optimal transference plan π in the case that the measures ν 1 and ν in 1.4 are absolutely continuous with respect to volume measure. Consider volume measure to be Riemannian volume measure here and in the sequel; the notion of absolute continuity is independent of the smooth Riemannian metric one chooses. We ll show that the π arises as the push-forward of ν 1 under a map defined by x x, F x where F : is a Borel map defined in terms of a potential function ϕ : R and the L-exponential map see Remark.8. The potential ϕ will arise via a Kantorovich dual formulation of the variational problem. Using this structure, we will be able to control π effectively. For example, π will be seen to give zero measure to the L-cut locus LCut τ1,τ which could be defined as the smallest subset of off which Q, τ 1 ;, τ is smooth. Developing this structure theory yields a Jacobian change of variables formula via Theorem.14 which will allow us later to effectively compute entropies of measures along L-Wasserstein geodesics, which are certain optimal paths of Borel probability measures on defined in terms of the L-exponential map and the potential ϕ mentioned above. Ultimately, the entropy calculations will be phrased in terms of L-Jacobi fields, which are analogues of Riemannian Jacobi fields in this setting. The necessary L-Jacobi field computations will be made in the next section. Virtually all of the material in this section is in one to one correspondence with the development of the standard theory of optimal transportation on manifolds by ccann [10] and Cordero-Erausquin, ccann and Schmuckenschläger [6]. As mentioned earlier, one could also appeal to [18]. We follow their route as closely as possible, and only give brief sketches of proofs where little adaptation is necessary. Our main goal here is to point out the exact analogues of their results in our setting. The first deviation of presentation of the Legendre-Fenchel-type transform used in the classical theory is motivated by the asymmetry of our cost function. Given a continuous function ϕ : R, we define the function ˆϕ : R by ˆϕy = inf x [Qx, τ 1; y, τ ϕx]..1 Likewise, if ψ : R is continuous, then ˇψ : R is defined by ˇψx = inf y [Qx, τ 1; y, τ ψy].. These transforms depend on τ 1 and τ, but those parameters can be viewed as fixed for now. Indeed, let us abbreviate Qx, y := Qx, τ 1 ; y, τ where no confusion will arise. As mentioned above, these transforms are the analogues of the c-transform in classical mass transportation see [10], for example with slightly different notation to emphasise the asymmetry of Q,. It is straightforward to check that taking one transform and then the other can only increase the original function: ˇˆϕ ϕ; ˆˇψ ψ..3 We have equality in, say, the first of these inequalities if ϕ = ˇψ for some ψ, because then ˇˆϕ = ˇˆˇψ ˇψ = ϕ. 7

8 Definition.1. Given a Ricci flow gτ, we call a function ϕ : R reflexive with respect to the interval [τ 1, τ ] if it is continuous, and satisfies ˇˆϕ = ϕ. This concept is called c-concavity in the classical theory of optimal transportation. Taking these transforms improves regularity in the following sense. The proof can be adapted from [10]. Lemma.. c.f.[10, Lemma ]. Suppose that there exists K < such that for all x, the Lipschitz constant of Qx, is no more than K. Then for all continuous ϕ : R, the function ˆϕ is also Lipschitz with Lipschitz constant no more than K. Here, Lipschitz is with respect to gτ. Similarly, Lipschitz control on Q, y gives Lipschitz control on ˇψ. Generally, we will not state similar results obtained by switching x and y. As we recall in Appendix A, Q is Lipschitz in both its variables. Throughout this section, we will be implicitly using the consequence of this lemma, via Rademacher s theorem, that any reflexive ϕ is differentiable almost-everywhere. We wish to work towards a Kantorovich dual formulation of the L-optimal transportation problem. Define Σ = {ϕ, ψ ϕ, ψ : R continuous and ϕx + ψy Qx, y x, y }, and, given Borel probability measures ν 1 and ν on, define J : Σ R by Jϕ, ψ = ϕ dν 1 + ψ dν. Lemma.3. c.f. [10, Proposition 3]. There exists a reflexive ϕ such that the supremum of J over Σ is attained at ϕ, ˆϕ. The proof following [10] is based on showing that if ϕ, ψ Σ, then ˇˆϕ, ˆϕ Σ and Jϕ, ψ J ˇˆϕ, ˆϕ. By virtue of Lemma., this allows one to alter any maximising sequence ϕ i, ψ i to one with controlled Lipschitz continuity, which enables us to pass to a limit via the Ascoli-Arzelà theorem to get a maximum. By definition of the transform.1, we have ϕx + ˆϕy Qx, y for all x, y. The case of equality is special: Lemma.4. c.f. [10, Lemma 7]. Suppose that ϕ is reflexive, and is differentiable at x. Then ϕx + ˆϕy = Qx, y if and only if y = L τ1,τ exp x ϕx..4 In this case, Q, y is differentiable at x, and ϕx = Q, yx. The gradient here is with respect to gτ 1. differentiability of ˆϕ at y. There is an analogous result in the case of Remark.5. Whenever we have x, y such that ϕx+ ˆϕy = Qx, y, the function ϕ must be a support function or lower barrier for Q, y ˆϕy near x. That is, the former function lies below the latter near x, with equality at x. This will repeatedly allow us to relate differentiability and convexity properties of ϕ and Q, y at such points x. 8

9 Concerning the proof of the lemma analogous to that in [10], for the only if part, note that by Remark.5, and the differentiability of ϕ at x, the function Q, y admits ϕx as a subgradient at x. oreover, by Lemma A.3 in Appendix A, Z is a supergradient of it at x, where Z Ωx, τ 1 ; τ T x satisfies y = L τ1,τ exp x Z. See Appendix A for notation. The supergradient and subgradient must then coincide as a genuine gradient, Z = ϕx. This is enough to establish.4. The if part is easier; by definition of ˆϕ, there always exists at least one point z at which ϕx + ˆϕz = Qx, z, and by what we have seen, this z must coincide with any y satisfying.4. These considerations put us in a position to construct maps F which transport certain measures in an optimal way. Theorem.6. c.f. [10, Theorem 8]. Suppose σ is a Borel probability measure which is absolutely continuous with respect to any volume measure on. Suppose that ϕ is a reflexive function. Then F :, a Borel map defined at points of differentiability of ϕ by F x := L τ1,τ exp x ϕx,.5 minimises the functional Qx, F x dσx amongst all Borel maps F such that F # σ = F # σ. Any other minimiser must agree with F σ-a.e. The proof follows exactly as in [10]: For any F as in the theorem, and any u, v Σ, then with J defined with respect to ν 1 = σ and ν = F # σ, we have Ju, v = u dσ+ v df # σ = u dσ+ v F x dσx Qx, F x dσx..6 But by Lemma.4 and the almost-everywhere differentiability of ϕ, we have ϕx + ˆϕF x = Qx, F x for almost all x with respect to any volume measure and hence Jϕ, ˆϕ = Qx, F xdσx. Combining with.6, we find that Jϕ, ˆϕ = sup J = inf Σ F Qx, F xdσx = Qx, F xdσx, and in particular, that F is the sought minimiser. If F is any other minimiser, we must still have ϕx + ˆϕ F x = Qx, F x for σ-almost all x, and by Lemma.4, we then know that F x = F x for σ-almost all x. Given the previous theorem, one would like to be able to find a reflexive ϕ and hence F to make the measure F # σ coincide with a measure of our choice: Theorem.7. c.f. [10, Theorem 9]. Suppose that ν 1 and ν are Borel probability measures, with ν 1 absolutely continuous with respect to any volume measure on. Then there exists a reflexive function ϕ : R such that Borel F : defined at points of differentiability of ϕ by.5 satisfies F # ν 1 = ν. The proof mimics that of [10, Theorem 9]. The function ϕ is that given by Lemma.3 9

10 Remark.8. Theorem.7 can be extended using Theorem.6 to assert that the optimal transference plan π in the definition 1.4 of V ν 1, τ 1 ; ν, τ is given by the push-forward of ν 1 under the map x x, F x. Returning to Lemma.4 and the definition of F from Theorem.6, we see that the image of F at a point x of differentiability for ϕ, is the unique point y at which ϕx + ˆϕy = Qx, y. Following [6], we now view F as the multi-valued function which assigns to an arbitrary point x the set of points y at which ϕx + ˆϕy = Qx, y. We will tend to abuse notation by occasionally retaining the old viewpoint for F at points of differentiability of ϕ. The following is merely a fragment of the proof of Lemma.4, but is included as the analogue of [6, Lemma 3.7], and is needed to prove Lemma.13 below. Again, the terminology Ωx, τ 1 ; τ comes from Appendix A. Lemma.9. If ϕ is reflexive, y F x and we pick Z Ωx, τ 1 ; τ T x such that y = L τ1,τ exp x Z, then Z is a supergradient of ϕ at x. We now turn to study second derivatives of Q and potentials ϕ. We are particularly interested in semiconcavity properties. If necessary, see Appendix A for the definition of semiconcave. Given any reflexive function ϕ, we can pick arbitrary points x and y F x and consider ϕ as a support function for Q, y ˆϕy at x as in Remark.5. This implies that for u T x sufficiently small, ϕexp x u + ϕexp x u ϕx u Qexp x u, y + Qexp x u, y Qx, y u,.7 where we are using the exponential map with respect to gτ 1. We then see that ϕ inherits the uniform semiconcavity of Q from Lemma A.4 in Appendix A, and we may deduce semiconcavity of ϕ from [6, Lemma 3.11]: Lemma.10. A reflexive function is semiconcave. We have already seen that a reflexive function ϕ is differentiable almost everywhere, because it is Lipschitz. By virtue of the semiconcavity of ϕ, we can be sure also that a Hessian in the sense of Alexandrov exists almost everywhere see [6], [1]. The following lemmata obtain refined control at points where this Hessian exists. See Appendix A for a discussion of the L-cut locus LCut and its subset LCut τ1,τ. Lemma.11. c.f. [6, Proposition 4.1a]. Suppose that ϕ : R is a reflexive function which admits a Hessian at x. With F x still defined by.5, we have x, τ 1 ; F x, τ / LCut hence Q, F x is smooth near x and at x there holds [Q, F x ϕ] = 0; Hess [Q, F x ϕ] 0..8 To prove this, we have, similarly to.7, that for u T x sufficiently small, ϕexp x u + ϕexp x u ϕx u Qexp x u, F x + Qexp x u, F x Qx, F x u, and since the left-hand side is controlled from below in the limit u 0 because the Hessian of ϕ exists the right-hand side must be also. By Lemma A.5 in Appendix A, this implies that x, τ 1, F x, τ / LCut hence the local smoothness of Q by Lemma 10

11 A. and.8 follows by returning to Remark.5 and using the fact that Q, F x ϕ has a minimum at x. The first part of.8 is already contained in Lemma.4. Combining Theorem.7 and Remark.8 with this lemma, we obtain: Corollary.1. Suppose that ν 1 and ν are Borel probability measures, with ν 1 absolutely continuous with respect to any volume measure on. If we denote by π the optimal transference plan in the definition 1.4 of V ν 1, τ 1 ; ν, τ, then πlcut τ1,τ = 0. We next want to define a differential df for the map F, where such a notion makes sense, and confirm that it has the properties one would expect given the name and notation. Lemma.13. c.f. [6, Proposition 4.1b]. Suppose that ϕ : R is a reflexive function which admits a Hessian at x. Define F again by.5, and a map df x : T x T F x by df x := 1 dl τ 1,τ exp x ϕx [hess Q, F x ϕ x],.9 where hessfx is the Hessian of a function f : R viewed as an endomorphism of T x i.e. the covariant derivative of the gradient of f. Then for u T x, we have sup v df xu = o u,.10 where the supremum is taken over all v T F x such that exp gτ F x and v gτ = df x, exp gτ F x v, τ. v F expgτ1 x u It is worth pointing out that when ϕ is smooth in a neighbourhood of x making F a smooth single-valued map in a neighbourhood of x then this formula for df x coincides with the differential of F as classically defined. As usual, the lemma above follows by adapting the corresponding proof from [6]. The same is true for the following result which uses the differential we have just defined to give a Jacobian identity. Theorem.14. c.f. [6, Theorem 4.]. Suppose that ν 1 and ν are Borel probability measures on, which are absolutely continuous with respect to any volume measure. Let f τ1 and f τ be the densities defined by dν 1 = f τ1 dµτ 1 and dν = f τ dµτ. If ϕ : R is a reflexive function for which F # ν 1 = ν where F is from.5 as provided by Theorem.7, then there exists a Borel set K with ν 1 K = 1 such that ϕ admits a Hessian at each x K; for all x K, we have f τ1 x = f τ F x det df x 0. We now have enough technology to construct L-Wasserstein geodesics, along the lines of [6, Section 5]. We define these to be one-parameter families of measures V τ, with τ [τ 1, τ ], which arise as the push-forwards F τ # ν 1 by Borel maps F τ : defined at points of differentiability of ϕ by F τ x := L τ1,τ exp x ϕx

12 The theory above involving F has always required that the function ϕ in the definition of F is reflexive, or more precisely, reflexive with respect to [τ 1, τ ]. In order to apply the theory we have developed to F τ as well as F = F τ, we must check that such a function ϕ is also reflexive with respect to [τ 1, τ]. Lemma.15. If ϕ : R is a reflexive function with respect to [τ 1, τ ], then for any τ τ 1, τ, it is also reflexive with respect to [τ 1, τ]. Proof. By definition of Q, we have Qa, τ 1 ; y, τ Qa, τ 1 ; z, τ + Qz, τ; y, τ, and so with respect to [τ 1, τ], [ ] ˇˆϕx = inf Qx, τ 1 ; z, τ inf [Qa, τ 1; z, τ ϕa] z a [ ] inf Qx, τ 1 ; z, τ + Qz, τ; y, τ inf [Qa, τ 1; y, τ ϕa].1 z a = Qx, τ 1 ; y, τ inf [Qa, τ 1; y, τ ϕa]. a If we now minimise over y, the right-hand side becomes precisely ˇˆϕx with respect to [τ 1, τ ], which is ϕx by hypothesis. Keeping in mind the first inequality of.3, the proof is complete. Note in particular, that in the context of Theorem.6, the maps F τ will all map σ optimally to V τ := F τ # σ. Lemma.16. c.f. [6, Lemma 5.3]. If ϕ : R is a reflexive function and F τ is defined as in.11, for x in the subset of of full measure on which ϕ is differentiable, then F τ is injective. In practice, we need a quantified version of this: Lemma.17. c.f. [6, Proposition 5.4]. Suppose that ν 1 and ν are Borel probability measures on which are both absolutely continuous with respect to any volume measure. If ϕ : R is a reflexive function such that F τ # ν 1 = ν, then for all τ τ 1, τ ], the interpolant measure V τ := F τ # ν 1 is also absolutely continuous with respect to any volume measure. Again, the proof is a translation of that in [6]. The analogue of the condition Hess d F t x tφ > 0 in that proof in [6] translates to HessQ, τ 1 ; F τ x, τ ϕ > 0 in our setting, whilst the inequality Hess d F t x t d F x 0 from [6] is simply HessQ, τ 1 ; F τ x, τ Q, τ 1 ; F τ x, τ 0 for us which follows immediately from the analogue of the triangle inequality Qa, τ 1 ; F τ x, τ Qa, τ 1 ; F τ x, τ + QF τ x, τ; F τ x, τ. In the next section, we want to analyse the behaviour of the classical entropy to be defined in 3.15 along L-Wasserstein geodesics see also [11], [6] and the references therein. We will compute this functional using the second part of Theorem.14 applied to F τ, and hence we need to compute det df τ x for x at certain points where ϕ admits a Hessian, where df τ x := 1 dl τ 1,τ exp x ϕx [hess Q, τ 1 ; F τ x, τ ϕ x], is the generalisation of.9. In practice, we will do that with the following observation c.f. [6] involving L-Jacobi fields which will be discussed further in Section 3. 1

13 Lemma.18. Suppose that ϕ : R is a reflexive function which admits a Hessian at x, and that Ŷ T x. Let γ : [τ 1, τ ] be the L-geodesic γτ = F τ x, and define Y Γγ T by Y τ := df τ xŷ for τ τ 1, τ ], and Y τ 1 := lim τ τ1 Y τ. Then Y τ is the L-Jacobi field along γ, with initial data Y τ 1 = Ŷ and D τ Y τ 1 = 1 τ 1 hessϕŷ. Remark.19. Because of this lemma, we find that if we choose any orthonormal basis {Ŷi} i=1,...,n for T x with respect to gτ 1 and consider the L-Jacobi fields Y i Γγ T determined by Y i τ 1 = Ŷi and D τ Y i τ 1 = 1 τ 1 hessϕŷi, then det df τ x = det Y i τ, Y j τ 1 gτ. 3 Behaviour of Boltzmann-Shannon entropy along L- Wasserstein geodesics In this section, we perform the computations for L-Jacobi fields which allow us to understand the behaviour of the entropy along an L-Wasserstein geodesic. The discussion at the end of the last section motivates the inequalities of the following lemma. We continue to consider a smooth reverse Ricci flow gτ defined on an open time interval including some interval [τ 1, τ ] with 0 < τ 1 < τ. Lemma 3.1. Suppose that γ : [τ 1, τ ] is an L-geodesic, and {Y i τ} i=1,...,n is a set of L-Jacobi fields along γ which form a basis of T γτ for each τ [τ 1, τ ], with {Y i τ 1 } orthonormal and D τ Y i, Y j symmetric in i and j at τ = τ 1. Then defining α : [τ 1, τ ] R by ατ = 1 ln det Y iτ, Y j τ gτ, and writing σ = τ 1, we have and d σα dσ where X = γ τ as before, and d α dσ = 4τ 1 d τ 1 dτ is the Hamilton Harnack quantity [9], [13]. dα τhx, 3.1 dτ = 4 d τ 3 dα τ 3 HX nτ 1, 3. dτ dτ HX := R τ XR + RicX, X R τ We clarify that Ric denotes the Ricci curvature of gτ viewed as a bilinear form, while Rc refers to that tensor viewed as an endomorphism, using gτ. Proof. The starting point for proving this is the equation for an L-Jacobi field Y τ D τ Y := D τ D τ Y = RX, Y + 1 Y R Y RcX RcD τ Y 1 τ D τ Y + X RcY [ Ric, X, Y ] #, 3.3 where we are using the sign convention RX, Y Z = X Y Z + Y X Z + [X,Y ] Z, and other conventions from [16]. 13

14 This equation looks at first glance somewhat different to the L-Jacobi equation elsewhere in the literature e.g. [5, 7.11] but our second derivative term is a little different to the conventional one, as we now clarify. Given a curve γ : [τ 1, τ ], and a metric g on, we denote by Dτ g the pull-back of the Levi-Civita connection of g by γ, acting in the direction τ. Given a flow of metrics e.g. a Ricci flow our previous notation D τ then coincides with Dτ gτ in this more general notation. At τ = ˆτ, the second derivative D τ D τ Y = D gτ τ D gτ τ Y = D gˆτ τ D gτ τ Y is then not equal to Dτ gˆτ Dτ gˆτ Y in general since the connection itself needs to be differentiated. Considering [16, Proposition.3.1], we have, at τ = ˆτ that D τ D τ Y = Dτ gˆτ Dτ gˆτ Y + Y RcX + X RcY [ Ric, X, Y ] #, 3.4 which accounts for the extra terms. Consider the frame field e i Γγ T, i = 1,..., n, with e i τ 1 := Y i τ 1 orthonormal, satisfying the ODE D τ e i + Rce i = Then {e i τ} is an orthonormal frame for all τ [τ 1, τ ]. We write Y j τ = A kj τe k τ for a τ-dependent n n matrix A. By applying D τ both once and twice, and taking the inner product gτ with e i, we find that and A ij = D τ Y j, e i + A kj Rice k, e i, 3.6 A ij = D τ Y j, e i + A kjrice k, e i + A kj D τ Rce k, e i. 3.7 The first term on the right-hand side of 3.7 can be dealt with using 3.3 and the definition of A ij. We find that [ Dτ Y j, e i = A kj RmX, e k, X, e i + 1 HessRe i, e k + X Rice i, e k ek RcX, e i ei RcX, e k + Rc e k, e i + 1 ] τ Rice i, e k A kjrice i, e k 1 τ A ij. 3.8 The inner product of the third term on the right-hand side of 3.7 can be expanded out, using the definition of {e i }, to give D τ Rce k, e i = Ric τ e i, e k + X Rice i, e k 3 Rc e k, e i. 3.9 One pitfall to avoid here is that while Ric and Rc differ only by raising/lowering an index, the tensors Ric τ and Rc τ do not, because raising/lowering an index involves using the metric gτ which depends on τ. Combining 3.7, 3.8 and 3.9, we find that A + 1 τ A = A, 3.10 where τ is the τ-dependent n n symmetric matrix given by ik = RmX, e i, X, e k + 1 HessRe i, e k ek RcX, e i ei RcX, e k + X Rice i, e k Rc e k, e i + 1 τ Rice i, e k + Ric τ e i, e k

15 The trace of is then tr = RicX, X + 1 R + δricx + XR Ric + R τ + tr Ric τ, but exploiting the contracted second Bianchi identity δric + dr = 0 using the notation and conventions of [16,.1.9] along with the evolution equation R τ = R + Ric for the scalar curvature [16, Proposition.5.4] and the fact that tr Ric τ = R τ + Ric from [16, Proposition.3.6] this simplifies to tr = 1 HX. 3.1 We are now in a position to compute the volume element ατ of the lemma, in the spirit of classical comparison geometry, and following the analogous [7, Lemma 6]. By definition, ατ = ln det A, and so dα dτ = tr da dτ A 1 and d α dτ = tr da da A 1 dτ dτ A 1 tr d A dτ A 1 If we define B := da dτ A 1, then this may be combined with 3.10 and 3.1 to give τ 1 d τ 1 dα da da d A = tr A 1 dτ dτ dτ dτ A 1 tr dτ + 1 da A 1 τ dτ = trb HX,. and τ 3 d τ 3 dα = tr B 1 dτ dτ τ I + 1 HX n 4τ It remains to show that B and hence also B 1 τ I is symmetric, so that the first terms on the right-hand sides of 3.13 and 3.14 are weakly positive. But following [7, Lemma 6] by writing B T B = A 1 T HA 1, where H := dat da dτ A AT dτ, and noting that d dτ τ 1 H = 0 and that Hτ 1 is the zero matrix because at τ = τ 1, A = I, and using 3.6 and the hypothesis of the lemma da dτ τ 1 is symmetric we see that Hτ is zero for any τ, and hence B is symmetric for any τ. We now turn to consider the Boltzmann-Shannon entropy of probability measures f dµ, where µ is Riemannian volume measure, and f is a suitably regular weakly positive function on, defined by Efdµ = f ln f dµ We are now in a position to investigate the behaviour of this entropy along L-Wasserstein geodesics as defined in the previous section with a result analogous to [11, Lemma 8]. We re-use the alternative variable σ = τ 1. Lemma 3.. Suppose that V τ is an L-Wasserstein geodesic, for τ [τ 1, τ ], induced by a potential ϕ : R, with V τ1 and V τ both smooth, and write dv τ = f τ dµτ where µτ is the volume measure of gτ. Then for all τ [τ 1, τ ], we have f τ L ln Lµτ, and the function τ EV τ is semiconvex and satisfies, for almost all τ [τ 1, τ ] where σ EV τ admits a second derivative in the sense of Alexandrov 4τ 1 d dτ τ 1 dev τ dτ = d dσ EV τ τ 15 HXτdV τ1 3.16

16 4 d τ 3 dτ dev τ = d dτ dσ σev τ τ 3 HXτdV τ1 nτ 1, 3.17 where Xτ, at a point x where ϕ admits a Hessian, is γ τ, for γ : [τ 1, τ ] the minimising L-geodesic from x to F x. oreover, the one-sided derivatives of EV τ at τ 1 and τ exist, with d ϕ dτ τ R, τ 1 + τ1 +EV, ln f τ1 dv τ τ 1 The ϕ of the lemma is the ϕ which induces the L-Wasserstein geodesic under consideration. This also induces a map F via.5 which we use below. Proof. The main ingredient in the proof is Lemma 3.1, applied to L-geodesics and L- Jacobi fields arising in Remark.19. Note that our volume density ατ will now have a suppressed x-dependency. At the core of the proof of Lemma 3. is the fact that we can relate the entropy at different values of τ in terms of the volume density α. With K τ the set provided by Theorem.14 with τ in place of τ, we have by that theorem f τ1 EV τ = ln f τ dv τ = ln f τ df τ # V τ1 = ln f τ F τ dv τ1 = ln dv τ1 K τ K τ det df τ = EV τ1 + ατdv τ1. K τ 3.19 We will combine this with Lemma 3.1 to yield the result. Indeed, that lemma gives immediately a lower bound d α dσ C, for C < independent of the point x at which we compute α, and this gives the semiconvexity of EV τ with respect to σ, or equivalently τ. For values of σ where σ EV τ admits a second derivative in the sense of Alexandrov, the identity d dσ EV EV σ+δ + EV σ δ EV σ α τ := lim δ 0 δ = σ σdv τ 1 follows from 3.19 thanks to Fatou s lemma, as in the proof of [11, Lemma 8]. This combines with Lemma 3.1 to give 3.16, and a similar approach gives By semiconvexity, the one-sided derivative of EV τ at τ = τ 1 must exist, allowing the possibility that it is. If we take any sequence τ k τ 1, and set K = k K τk, then we may exploit 3.19 once again to give that d dτ τ = lim τ1 +EV k K ατ k ατ 1 τ k τ 1 dv τ1. If α were a convex function of τ for each x, then the monotone convergence theorem would tell us that d dτ τ = α τ1 +EV τ 1 dv τ1, 3.0 and it is not hard to see that the same conclusion follows from the known semiconvexity of α. By definition of α, keeping in mind that the L-Jacobi fields on which α depends K 16

17 were chosen as in Remark.19, we have at τ = τ 1 α τ 1 = 1 [ ] d tr dτ Y i, Y j gτ = 1 tr [RicY i, Y j + D τ Y i, Y j + Y i, D τ Y j ] = R D τ Y i, Ŷi i 3.1 = R + 1 τ 1 ϕ. Returning to 3.0, and exploiting the semiconvexity of ϕ to be sure that the singular part of the distributional Laplacian D ϕ of ϕ is weakly positive, we have d dτ τ = Rx, τ 1 + τ1 +EV 1 K ϕ dv τ1 Rx, τ τ 1 D ϕ dv τ1. τ 1 3. Because dv τ1 = f τ1 dµτ 1, this gives 3.18 as desired. We have given 3.16 for use in future work. Here we only require 3.17, and then only the version of it one obtains by integrating with respect to τ not σ. Indeed, by semiconcavity of σev τ, we can see that 3.17 holds in the distributional sense, and integrates to [ τ 3 ] τ dev τ τ dτ τ 1 τ 1 = τ 3 HXτdV τ1 n τ 1 dτ Kx, τ 1, y, τ dπx, y n τ τ 1, 3.3 where K is defined in A.8 of Appendix A, and π is the push forward of V τ1 x x, F x as in Remark.8. by the map eanwhile, by exploiting Lemma.11 to write ϕx = 1 Qx, τ 1 ; F x, τ for appropriate x, where 1 Q represents the gradient of Q with respect to its first argument, and with respect to gτ 1, we can rewrite 3.18 as d dτ EV τ τ1 Rx, τ Qx, τ 1 ; y, τ τ 1, ln f τ1 x dπx, y, 3.4 where π is again the push forward of V τ1 by the map x x, F x. Taking this viewpoint, we have the right notation to give the analogous inequality for the other one-sided derivative: d Qx, τ 1 ; y, τ dτ EV τ Ry, τ + τ, ln f τ y dπx, y, 3.5 τ where Q is the gradient with respect to the y argument, and with respect to gτ. Combining 3.3, 3.4 and 3.5, we obtain the following corollary which is what we shall require in the proof of Theorem

18 Corollary 3.3. Under the hypotheses of Lemma 3., we have 3 K τ 1 Rx, τ1 τ 1 1 Q, ln f τ1 x + τ 3 gτ1 Ry, τ τ Q, ln f τ y gτ dπx, y 3.6 n τ τ 1, where π is the optimal transference plan from Vτ 1 to Vτ for L-optimal transport. 4 Proof of Theorem 1.1 We will follow [11, Section 4] as closely as possible. All we need to show is that d + Θ Θs Θ0 ds := lim sup 0. s=0 s 0 s Defining hs := V ν 1 τ 1 s, τ 1 s; ν τ s, τ s, this is equivalent to proving that d + h ds 1 s=0 h0 + n τ τ Let us write dν i τ = u i τdµτ, so u i τ : 0, is the probability density of ν i τ, for i = 1,. Since ν 1 and ν are diffusions, by Fourier s law, if we take families ψi τ : of diffeomorphisms with ψ τi i the identity, generated by ln u i, then ψi τ #ν i τ i = ν i τ for τ near τ i. If we now let π 0 be the minimiser for the variational problem 1.4 in the case of V ν 1 τ 1, τ 1 ; ν τ, τ, and use π s := ψ τ1s 1 ψ τs # π 0 as a competitor for the variational problem defining V ν 1 τ 1 s, τ 1 s; ν τ s, τ s, then we may compute similarly to [11] that hs h0 Qψ τ1s 1 x, τ 1 s; ψ τs y, τ s Qx, τ 1 ; y, τ dπ 0 x, y, and hence keeping in mind Lemma A. and Corollary.1 we have 4. d + h ds s=0 = d ds Q ψ τ1s 1 x, τ 1 s; ψ τs y, τ s dπ 0 x, y s=0 1 Q, ln u 1 τ 1 τ 1 + Q τ 1 + Q, ln u τ τ + Q τ τ 1 τ 4.3 where the inner products used are gτ 1 and gτ respectively. Exploiting Lemma A.6 from Appendix A, this may be written d + h ds s=0 τ 1 1 Q, ln u 1 τ 1 τ Q, ln u τ τ Ry, τ τ 1 Rx, τ1 + K dπ h0. By Corollary 3.3, we deduce our desired 4.1. dπ 0, 18

19 A Appendix Theory of L-length In this appendix we briefly survey the theory of Perelman s L-length, and the distance Q it induces. Some of this material can be found in the foundational paper [13, 7], described for Ly, τ := Qx, 0; y, τ. Details of the remaining parts can either be found in [5, Chapter 7] or [19], or can be arrived at by adapting the analogous theory of Riemannian length in Riemannian geometry. Throughout this appendix, we will be considering a smooth reverse Ricci flow gτ on a closed manifold, defined on an open time interval ˆτ 1, ˆτ where 0 < ˆτ 1 < τ 1 < τ < ˆτ. We use the notation Υ := {x, τ a ; y, τ b x, y and ˆτ 1 < τ a < τ b < ˆτ }. We will continue to abbreviate Qx, τ 1 ; y, τ by Qx, y. We have already recalled, in the introduction and Section, the definitions of L-length, L-geodesics and the L-exponential map, and we shall require these in this appendix. One can also define an L-exponential map backwards in time. The notion of L-geodesic also induces a notion of L-Jacobi field, and we refer the reader to [5, Chapter 7] for details. The first point to note is that the distance function Qx, τ 1 ; y, τ defined in the introduction is locally Lipschitz on Υ, where we use the metric on Υ arising as the sum of gτ 1, dτ 1, gτ and dτ corresponding to the four respective arguments of Q. This follows by a variation on the argument for showing that Perelman s L distance is locally Lipschitz see e.g. [5, Lemma 7.30] or [19]. In particular, for fixed τ 1 and τ, the function Q, τ 1 ; y, τ is Lipschitz with Lipschitz constant independent of y. Just as for Riemannian distance, Q need not be smooth; Q will fail to be differentiable on some subset LCut Υ which we will define and study now. It is convenient to define Ωx, τ 1 ; τ := {Z T x γ : [τ 1, τ ] defined by γτ = L τ1,τ exp x Z is a minimising L-geodesic}. A.1 This set clearly shrinks as τ is increased, and exhausts T x in the limit τ τ 1. Beware, however, that it need not be star-shaped as would its classical Riemannian analogue. Define Ω x, τ 1 ; ˆτ to be the intersection of all the sets Ωx, τ 1 ; τ, over τ τ 1, ˆτ. For Z T x \Ω x, τ 1 ; ˆτ, define τx, τ 1 ; Z := sup{τ τ 1, ˆτ Z Ωx, τ 1 ; τ} τ 1, ˆτ. We can then define the possibly empty set LCut := {x, τ 1 ; L τ1, τx,τ 1;Z exp x Z, τx, τ 1 ; Z x, τ 1 ˆτ 1, ˆτ, Z T x \Ω x, τ 1 ; ˆτ }. A. It will also be convenient to define the slice LCut τ1,τ to be the subset of LCut consisting of those points of the form x, τ 1 ; y, τ for some x, y. Remark A.1. uch of the classical theory of cut loci in Riemannian geometry carries over to LCut. In particular, LCut can be characterised as the union of two sets: the first consisting of points x, τ 1 ; y, τ such that there exists more than one minimising L-geodesic γ : [τ 1, τ ] with γτ 1 = x and γτ = y, and the second consisting of points x, τ 1 ; y, τ such that y is conjugate to x with respect to L-Jacobi fields along a minimising L-geodesic γ : [τ 1, τ ] with γτ 1 = x and γτ = y. This characterisation is used to prove the first three parts of the following lemma, using the techniques of the proof of [11, Lemma 5]. Implicit here is the existence of minimising L-geodesics between given end-points see, for example, [5, Lemma 7.7]. 19

20 Lemma A.. We have that: i The set LCut is closed in Υ. ii The function Q is smooth on Υ\LCut. iii The minimising L-geodesic corresponding to each point in Υ\LCut is smoothly dependent on that point in the sense that if we associate to each point x, τ 1 ; y, τ Υ\LCut the vector Z Ωx, τ 1 ; τ T x for which L τ1,τ exp x Z = y, then Z depends smoothly on x, τ 1 ; y, τ. iv On Υ\LCut we have Q τ 1 x, τ 1 ; y, τ = τ 1 Z Rx, τ 1 ; 1 Qx, τ 1 ; y, τ = Z, A.3 τ 1 where 1 Q denotes the gradient with respect to the first argument, x, using the metric gτ 1. Analogous formulae hold for the derivatives with respect to y and τ. The equations of part iv are similar to Perelman s formulae for the derivatives of L, from [13, 7]. When we write Z here, we mean its length with respect to gτ 1. To extend the lemma above, we need further notation. Suppose x, τ 1 ; y, τ Υ\LCut, and let γ : [τ 1, τ ] be the minimising L-geodesic from x to y. We write Xτ = γ τ, so τ 1 Xτ 1 coincides with the Z of the previous lemma. In place of A.3 we have Q τ 1 x, τ 1 ; y, τ = τ 1 Xτ1 Rx, τ 1 ; 1 Qx, τ 1 ; y, τ = τ 1 Xτ 1, and the corresponding formulae for the other derivatives of Q are then Q τ x, τ 1 ; y, τ = τ Ry, τ Xτ ; A.4 Qx, τ 1 ; y, τ = τ Xτ. A.5 We now have enough control on Q off LCut, but we need at least some control on Q across its whole domain. Up to now, we know simply that it is locally Lipschitz. The first observation is that although Q, y need not be everywhere differentiable on, it does admit a supergradient everywhere: Lemma A.3. For all x, y, if we pick Z Ωx, τ 1 ; τ T x such that y = L τ1,τ exp x Z, then Z is a supergradient of Q, y at x. One can easily construct an upper barrier for Q, y at x which implies this lemma, either using the so-called Calabi trick, or by considering the L-lengths of a smooth variation of a minimising L-geodesic from x to y. See [5, Lemma 7.3] for this latter approach to prove the corresponding result for L instead of Q. We now turn to look at second derivative properties of Q. We need several times in the paper that Q, y is semiconcave. This means that near each point, one can add a smooth function to give a geodesically concave function. This notion is independent of the choice of metric [1]. However, we also need Q, y to be uniformly semiconcave see [6] which is stronger and is the content of the following lemma. Lemma A.4. c.f. [6, Corollary 3.13]. There exists C such that for all x, y and v T x, lim sup r 0 Qexp gτ1 x rv, y + Qexp gτ1 x rv, y Qx, y r C. A.6 0

21 We stress that C here is independent of x and y and v. Note that according to [6, Lemma 3.11], such a uniform estimate implies semiconcavity. This uniform estimate can be proved by exploiting the second variation calculations of [13, 7]. See also the discussion of Hessian bounds in [19]. Although Lemma A.4 gives an upper bound for the left-hand side of A.6, there is no lower bound in general. Lemma A.5. c.f. [6, Proposition.5]. For each point x, τ 1 ; y, τ LCut, the function Q, τ 1 ; y, τ is not differentiable at x, and its Hessian is unbounded below in the sense that lim inf u 0 Qexp gτ1 x u, y + Qexp gτ1 x u, y Qx, y u =. A.7 To prove this, one can follow the proof of the corresponding Riemannian result from [6]. One should deal with each point of LCut differently depending on its place in the characterisation of LCut mentioned in Remark A.1. For example, if there are two distinct minimising L-geodesics from x to y along which x and y are not conjugate, then each can be used to construct upper barriers for Q, y with different gradients at x, and the result is clear in that case. When x and y are conjugate along a minimising L-geodesic, then one considers the L-index form refer to [5, 7.134] analogously to the proof of [6, Proposition.5]. Finally, the proof of the main theorem in Section 4 will require a formula involving the derivatives of Q which we derive now. Suppose x, τ 1 ; y, τ Υ\LCut, let γ : [τ 1, τ ] be the minimising L-geodesic from x to y, and write Xτ = γ τ as before. Following [13], we define K = Kx, τ 1, y, τ := τ τ 1 τ 3 HXτdτ, A.8 where H is the Hamilton Harnack quantity defined in Lemma 3.1. Perelman was only considering the case τ 1 = 0 at this point in his work, but the direct analogue of [13, 7.4] is 3 τ Ry, τ + Xτ 3 τ 1 Rx, τ1 + Xτ 1 = Kx, τ 1, y, τ + 1 Qx, τ 1; y, τ. A.9 Combining with A.4 and A.5, we find corresponding to [13, 7.5] the following lemma. Lemma A.6. Under Ricci flow, Q satisfies Q Q 3 3 τ + τ 1 = τ Ry, τ τ 1 Rx, τ1 + K 1 Q. A.10 τ τ 1 B Appendix Wasserstein and L-Wasserstein distance, and their infinitesimal versions Throughout this appendix, we will be considering a reverse Ricci flow gτ defined on a closed manifold, with τ ranging over an open interval containing [τ 1, τ ], for 0 < τ 1 < τ. We will be concerned with the limit in which τ 1 and τ approach each other. 1